geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the gyroelongated pentagonal pyramid is one of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s (). As its name suggests, it is formed by taking a

pentagonal pyramid In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual. The ''regular'' pentagonal pyramid has a base that is a regu ...

and "gyroelongating" it, which in this case involves joining a pentagonal

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

to its base. It can also be seen as a diminished icosahedron, an

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

with the top (a

, ) chopped off by a plane. Other Johnson solids can be formed by cutting off multiple pentagonal pyramids from an icosahedron: the

pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...

and

metabidiminished icosahedron In geometry, the metabidiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing two pentagonal pyramids () from a regular icosahedron, replacing two sets of five triangular faces of the ic ...

(two pyramids removed), and the

tridiminished icosahedron In geometry, the tridiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing three pentagonal pyramids () from a regular icosahedron, which replaces three sets of five triangular faces fro ...

(three pyramids removed).

Dual polyhedron

The dual of the gyroelongated pentagonal pyramid has 11 faces: 5 kites, 1 regular pentagonal and 5 irregular pentagons.

External links

* Johnson solids Pyramids and bipyramids {{Polyhedron-stub